Very good explanation of both transformations:

1. MA(1) == AR(inf) https://www.youtube.com/watch?v=QU_VNu3rJKY&list=PLUTpaEbqzDRy2vgDyzHJwvab2RUzVoS-o&index=16
2. MA(inf) == AR(1) https://www.youtube.com/watch?v=q0vz7dGlZL0&list=PLUTpaEbqzDRy2vgDyzHJwvab2RUzVoS-o&index=18

Main (and single) requirement for time series in form $c_t = f * c_{t-1} + e_t$ is: $|f| < 1$